A wide range of applications involve interface problems. In most of the cases, mathematical modeling of these interface problems leads to partial differential equations with non-smooth or discontinuous inputs and solutions, especially across material interfaces. Different numerical methods have been developed to solve these kinds of problems and handle the non-smooth behavior of the input data and/or the solution across the interface. The main focus of our work is the immersed finite element method to obtain optimal numerical solutions for interface problems. In this thesis, we present piecewise quadratic immersed finite element (IFE) spaces that are used with an immersed finite element (IFE) method with interior penalty (IP) for solving two-dimensional second-order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. An analysis of the constructed IFE spaces and their dimensions is presented. Shape functions of Lagrange and hierarchical types are constructed for these spaces, and a proof for the existence is established. The interpolation errors in the proposed piecewise quadratic spaces yield optimal <i>O</i>(h³) and <i>O</i>(h²) convergence rates, respectively, in the L² and broken H¹ norms under mesh refinement. Furthermore, numerical results are presented to validate our theory and show the optimality of our quadratic IFE method. Our approach in this thesis is, first, to establish a theory for the simplified case of a linear interface. After that, we extend the framework to quadratic interfaces. We, then, describe a general procedure for handling arbitrary interfaces occurring in real physical practical applications and present computational examples showing the optimality of the proposed method. Furthermore, we investigate a general procedure for extending our quadratic IFE spaces to <i>p</i>-th degree and construct hierarchical shape functions for <i>p</i>=3. / Ph. D.

In this thesis we present a parallel implementation of an engineering code which simulates the deformations caused when forces are applied to a piezoelectric-controlled smart structure. The parallel simulation, whose domain decomposition relies on the finite element representation of the structure, is created with an emphasis on scalability of both memory requirements and run time. We take into consideration sequential performance enhancements, the structure of a banded symmetric positive definite linear system, and the overhead required to completely distribute the problem across the processors. The resulting code is scalable, with the exception of a banded Cholesky factorization, which does not fully utilize the parallel environment. / Master of Science

A racing vehicle suspension system is a kinematic linkage that supports the vehicle under complex loading scenarios. The suspension also defines the handling characteristics of the vehicle. Understanding the loads that the suspension carries in a variety of loading scenarios is necessary in order to properly design a safe and effective suspension system. In the past, the Formula SAE team at Virginia Tech has used simplified calculations to determine the loads expected in the suspension members. This approach involves several large assumptions. These assumptions have been used for years and the justification for them has been lost. The goal of this research is to determine the validity of each of the assumptions made in the method used for calculating the vehicle suspension loads by hand. These assumptions include modeling the suspension as pinned-pinned truss members to prevent bending, neglecting any steering angle input to the suspension, and neglecting vertical articulation of the system. This thesis presents an approach to modeling the suspension member loads by creating a finite element (FE) model of the entire suspension system. The first stage of this research covers the validation of the current calculation methods. The FE model will replicate the suspension with all of the current assumptions and the member loads will be compared to the hand calculations. This truss-element-based FE model resulted in member loads identical to the hand calculations. The next stage of the FE model development converts the truss model to beam elements. This step is performed to determine if the assumption that bending loads are insignificant is a valid approach to calculating member loads. In addition to changing the elements used from truss to beam element, the suspension linkage was adapted to more accurately model the methods by which each member is attached to the others. This involves welding the members of each control arm together at the outboard point as well as creating a simplified version of the pull rod mounting bracket on the upper control arm. The pull rod is the member that connects the ride spring, damper, and anti-roll bar to the wheel assembly and had previously been mounted on the upright. This model reveals reduced axial components of load but increases in bending moments sizable enough to reduce the resistance to buckling of any member in compression. The third stage of model development incorporates the steer angle that must be present in loading scenarios that involve some level of cornering. An analysis of the vehicle trajectory that includes the effects of slip angle is presented and used to determine the most likely steer angle the vehicle will experience under cornering. The FE model was adapted to include the movement of the steering linkage caused by driver input. This movement changes the angle of the upright and steering linkage as well as the angle at which wheel loads are applied to the suspension. This model results in a dramatic change in member loads for loading cases that involve a component of steering input. Finally, the FE model was further enhanced to account for vertical movement of the suspension as allowed by the spring and damper assembly. The quasi-static loading scenarios are used to determine any member loading change due to vertical movement. The FE model is also used to predict the amount of vertical movement expected at the wheel center. This data can be used by the suspension designer to determine if changes to the spring rate or anti-roll bar stiffness will result in a more desirable amount of wheel movement for a given loading condition. This model shows that there is no change in the member loads due to the vertical movement of the wheel. This thesis concludes by presenting the most important changes that must occur in member load calculations to determine the proper suspension loading under a variety of loading scenarios. Finally, a discussion of future research is offered including the importance of each area in determining suspension loads and recommendations on how to perform this research. / Master of Science

Energy release rates are extracted from non-linear finite element analyses of a thin film bonded to a rigid substrate that is shaft-loaded at its free edge. This geometry is of interest because it simulates a probe test that has proven to be useful in characterizing the adhesion of thin, microelectronic coatings bonded to silicon wafers. Preliminary experimental results indicate that out-of-plane rather than in-plane loading dominates failure in the system. This work therefore focuses on out-of-plane film loading. To validate finite element and energy release rate methodologies, energy release rates from finite element analyses of pressurized and shaft-loaded blister tests are first correlated to theoretical limit cases. Upon validation, mode I, mode II, and mode III energy release rates are extracted from three-dimensional continuum finite element models of the edge-loaded thin film by a three-dimensional modified crack closure method. Having assumed a circular debond as observed experimentally, energy release rates are determined by a step-wise approach around the circumference. The progression of debond is simulated in multiple analyses by altering the boundary conditions associated with increasing the debond radius. Mechanical loading is supplemented with thermal loading, introducing residual stresses in the non-linear analyses. A sensitivity analysis of energy release rates to residual stress is performed. The results indicate that inclusion of residual stress has an important role in both the magnitude and mode-mixity of energy release rates in the thin film. Increasing the length of debond effectively transitions the film from a shearing mode to a bending mode, thereby significantly impacting each mode of energy release rate differently. / Master of Science

The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the problem of proving global existence and uniqueness is more difficult and fails even for some very simple ordinary differential equations. It is still not known if the 3D Navier-Stokes equation have global unique solutions and this open problem is one of the Millennium Prize Problems. However, many of these mathematical models are extremely useful in the understanding of complex physical systems. For years people have considered methods for modifying these equations in order to obtain models that still capture the observed fundamental physics, but for which one can rigorously establish global results. In this thesis we focus on a taming method to achieve this goal and apply taming to modeling and numerical problems. The method is also applied to a class of nonlinear differential equations with conservative nonlinearities and to Burgers’ Equation with Neumann boundary conditions. Numerical results are presented to illustrate the ideas. / Master of Science

The advent of finite element methods has changed the tire industry's design process over the past three decades. Analyses, previously impractical using analytical methods and physically limited by experimental methods, can now be performed using computational methods. This decreases the cost and time associated with bringing a new design to the marketplace; however some physical testing is still required to validate the models. The design, fabrication, installation, and operation of a tire, suspension, and chassis test fixture (TiSCTeF) is detailed as part of this study. This fixture will support the validation of effective, parametric finite element models currently under development, as well as the design and testing of suspension and chassis components for the Virginia Tech Formula SAE team. The fixture is designed to use the Formula SAE race car as the test platform. Initially, the fixture is capable of performing static load-deflection and free-rolling tire tests. Provision has been made in the design for incremental upgrades to support cornering tests and additional instrumentation. An initial load-deflection test has proven that the fixture is capable of creating reproducible data sets. Specific recommendations are made concerning the improvement of data quality for future tests. This study also presents a process for analyzing existing tire cornering data and eliminating anomalies to improve the effectiveness of normalization techniques found in the literature. The process is shown to collapse tire cornering data, which is partially ill- conditioned, onto master curves that consistently display the effect of inclination angle and tire inflation pressure on tire response. / Master of Science

[pt] Pela observação e análise das condições dos locais atingidos por violentos sismos destrutivos, pode-se demostrar a grande importância do comportamento do maciço de solo na resposta dinâmica das estruturas na superfície. Embora esta importância tenha sido há muito reconhecida, só recentemente uma investigação sistemática dos fatores intervenientes na interação dinâmica solo-estrutura vem merecendo a atenção adequada. Graças ao avanço de técnicas numéricas e métodos de análise, é possível hoje estudar-se este problema de forma eficiente, ampla e racional. O presente trabalho consiste em uma análise pelo método dos elementos finitos do problema do comportamento dinâmico dos tanques de armazenamento de combustível da Refinaria Costarricense de Petróleo (RECOPE) freqüentemente sujeitos a abalos sísmicos. A avaliação da amplificação sísmica dos solos da região da refinaria, o comportamento estrutural dos tanques sob solicitações dinâmicas, a análise comparativa dos métodos de projeto recomendados na Costa Rica, os efeitos da interação dinâmica solo-tanque, etc., são alguns dos pontos examinados em detalhe neste trabalho. O conhecimento do comportamento dinâmico dos tanques da Refinaria de RECOPE, Costa Rica, foi significativamente aperfeiçoado pela realização deste trabalho. / [en] Through observation and analysis of the local conditions of regions subjected to violent seismic excitations, one can conclude the great importance of the dynamic behavior of the soil mass on the response of structural systems founded in it. Although the importance of the soil behavior has long been recognized in the literature, systematic investigations on the influencing factors on the dynamic soil-structure interaction problems have received the due attention only recently.Given the modern developments in numerical methods as well as the now common availability of powerful microcomputers in design offices, it is now possible to study this problem following a rational and efficient manner.In the present work, an analysis of the dynamic behavior of steel tanks for fuel storage at the industrial plant of RECOPE (Petroleum Refinery of Costa Rica), frequently subjected to severe earthquakes, has been carried out using the finite element method.The main points investigated in this thesis are: the evaluation of dynamic amplifications due to local soil properties, the structural behavior of tanks under dynamic forces, the soil-structure interaction effects as well as a comparative study with the design procedures currently employed in Costa Rica.The technical knowledge about the dynamic behavior of RECOPE`s fuel tanks has been significantly improved by this research. / [es] Por la observación y análisis de las condiciones de los locales castigados por violentos sismos destructivos, se puede demostrar la gran importancia del comportamiento del macizo de suelo en la respuesta dinámica de las extructuras en la superficie. Aunque esta importancia há sido reconocida hace mucho tiempo, solo recientemente una investigación sistemática de los factores que intervienen en la interacción dinámica suelo extructura ha merecido atención adecuada. Gracias al avance de técnicas numéricas y métodos de análisis, es posible hoy estudiar este problema de forma eficiente, amplia y racional. EL presente trabajo consiste en un análisis por el método de los elementos finitos del problema del comportamiento dinámico de los tanques de almacenamiento de combustible de la Refinaría Costarricense de Petróleo (RECOPE) frecuentemente sujetos a sacudidas sísmicas. La evaluación de la amplificación sísmica de los suelos de la región de la refinaria, el comportamiento extructural de los tanques bajo solicitaciones dinámicas, el análisis comparativo de los métodos de proyecto recomendados en la Costa Rica, los efectos de la interacción dinámica suelo-tanque, etc., son algunos de los puntos examinados en detalle en este trabajo. EL conocimiento del comportamiento dinámico de los tanques de la Refinaría de RECOPE, Costa Rica, fue significativamente perfeccionado por la realización de este trabajo.

Hydrocarbon production or injection of fluids in the reservoir can produce changes in the rock stresses and in-situ geomechanics, potentially leading to compaction and subsidence with harmful effects in wells, cap-rock, faults, and the surrounding environment as well. In order to tackle these changes and their impact, accurate simulations are essential. The Mortar Finite Element Method (MFEM) has been demonstrated to be a powerful technique in order to formulate a weak continuity condition at the interface of sub-domains in which different meshes, i.e. non-conforming or hybrid, and / or variational approximations are used. This is particularly suitable when coupling different physics on different domains, such as elasticity and poroelasticity, in the context of coupled flow and geomechanics. In this dissertation, popular Domain Decomposition Methods (DDM) are implemented in order to carry large simulations by taking full advantage of current parallel computer architectures. Different solution schemes can be defined depending upon the way information is exchanged between sub-domain interfaces. Three different schemes, i.e. Dirichlet-Neumann (DN), Neumann-Neumann (NN) and MFEM, are tested and the advantages and disadvantages of each of them are identified. As a first contribution, the MFEM is extended to deal with curve interfaces represented by Non-Uniform Rational B-Splines (NURBS) curves and surfaces. The goal is to have a more robust geometrical representation for mortar spaces, which allows gluing non-conforming interfaces on realistic geometries. The resulting mortar saddle-point problem will be decoupled by means of the DN- and NN-DDM. Additionally, a reservoir geometry reconstruction procedure based on NURBS surfaces is presented as well. The technique builds a robust piecewise continuous geometrical representation that can be exploited by MFEM in order to tackle realistic problems, which is a second contribution. Tensor product meshes are usually propagated from the reservoir in a conforming way into its surroundings, which makes non-matching interfaces highly attractive in this case. In the context of reservoir compaction and subsidence estimation, it is common to deal with serial legacy codes for flow. Indeed, major reservoir simulators such as compositional codes lack parallelism. Another issue is the fact that, generally speaking, flow and mechanics domains are different. To overcome this limitation, a serial-parallel approach is proposed in order to couple serial flow codes with our parallel mechanics code by means of iterative coupling. Concrete results in loosely coupling are presented as a third contribution. As a final contribution, the DN-DDM is applied to couple elasticity and plasticity, which seems very promising in order to speed up computations involving poroplasticity. Several examples of coupling of elasticity, poroelasticity, and plasticity ranging from near-wellbore applications to field level subsidence computations help to show that the proposed methodology can handle problems of practical interest. In order to facilitate the implementation of complex workflows, an advanced Python wrapper interface that allows programming capabilities have been implemented. The proposed serial-parallel approach seems to be appropriate to handle geomechanical problems involving different meshes for flow and mechanics as well as coupling parallel mechanistic codes with legacy flow simulators. / text

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